In a large skillet, melt the butter over medium heat, pour off about 3 tablespoons into a small bowl, and set aside. Add the sausage to the skillet and cook, crumbling with a wooden spoon, until no longer pink, about 3 minutes.

Add the onion and celery and cook, stirring occasionally, until the onion is slightly wilted, about 6 minutes. Add the apples, pecans, thyme and sage, and stir to coat everything with the butter.

Add the chicken broth and bring to a simmer. Simmer until the apples are just tender, about 8 minutes. Scrape into a large bowl. In a small bowl, beat the egg with the half-and-half. Add the bread to the bowl with the sausage mixture and pour the egg mixture over it. Season with salt and pepper and toss well to moisten all of the bread, adding a little more broth if necessary.

Spread in the prepared baking dish, drizzle with the remaining melted butter and tent with foil (don’t let the foil touch the stuffing). Bake until heated through, about 20 minutes. Uncover and bake until the top is browned and crisp, about 30 minutes more.